In Representation Theory, the theme of the existence of a canonical basis has been explored quite a lot. I will limit myself in this question to the kind of canonical bases that arise from the Geometric Satake Correspondence (due to Mirkovic-Vilonen, followed up in works by Anderson, Kamnitzer and many others). Here, one studies certain sheaves on the Affine Grassmannian for the group G and from this geometrical setup, one gets a canonical basis for a finite dimensional irrep of the Langlands dual group $G^\vee$.

Now, to someone who has studied the theory of finite dimensional representations of compact Lie Groups (Ex : can do tensor products, find branching laws etc, maybe not by the most efficient means, but by some means), is there any striking property of this theory that one can take as a hint of existence of the kind of canonical basis recalled above ?

Now, let me note here that if one takes the existence of the canonical basis, then one can prove non-trivial facts about finite dimensional representation theory in a nice way. For example Anderson (in this paper) proves a relationship between weight multiplicities and tensor product multiplicities using the MV basis. But, I don't think one can turn this argument in the other direction. That is, the existence of a relation between weight multiplicites and tensor product multiplicities is, by itself, not enough to indicate the existence of a canonical bases. I am looking for something in this direction.